Mixed Partially Hyperbolic System

• Mixed partially hyperbolic systems are diffeomorphisms on compact manifolds with an invariant dominated splitting, combining robust hyperbolic behavior and mixed center dynamics.
• They feature uniform expansion/contraction in stable and unstable bundles while the center bundles display nonuniform, mixed Lyapunov exponent profiles.
• These systems possess finitely many ergodic SRB measures with basins covering full Lebesgue measure, ensuring statistical stability under perturbations.

A mixed partially hyperbolic system is a diffeomorphism (or flow) on a compact manifold that admits a dominated splitting of the tangent bundle into invariant subbundles reflecting both hyperbolic (expanding/contracting) and center (intermediate or "mixed") behaviors, with structural properties and statistical consequences sharply distinct from classical uniformly hyperbolic (Anosov) settings. The key feature is the presence of at least one "center" direction exhibiting non-uniform and genuinely mixed (coexisting expanding and contracting) Lyapunov exponent profiles for physically relevant invariant measures.

1. Formal Definition and Structure

Let $f\colon M\to M$ be a $C^{1+}$ (at least $C^2$ for SRB theory) diffeomorphism of a compact Riemannian manifold $M$. A mixed partially hyperbolic system is characterized by a $Df$–invariant dominated splitting of the tangent bundle,

$$TM = E^{uu} \oplus_{\succ} E^{cu} \oplus_{\succ} E^{cs}$$

or, in a more classical form,

$$TM = E^{s} \oplus_{\succ} E^{c} \oplus_{\succ} E^{u}$$

where $E^{uu}$ and $E^{cs}$ are strongly unstable and strongly stable bundles (with uniform exponential growth/decay), while $E^{cu}$ and $E^{c}$ are "center" or "center-unstable" bundles displaying neither uniform contraction nor expansion. The domination notation $\oplus_{\succ}$ signifies strict inequalities for the rates of expansion/contraction, i.e., for all $x \in M$ and $n \ge 1$,

$$\|Df^n|_{E^{cs}(x)}\| \le c \sigma^n, \quad m(Df^n|_{E^{uu}(x)}) \ge c^{-1}\nu^n,$$

and

$$\|Df^n|_{E^{cs}(x)}\| < m(Df^n|_{E^{cu}(x)}) \leq \|Df^n|_{E^{cu}(x)}\| < m(Df^n|_{E^{uu}(x)}).$$

In the mixed regime, the center bundle itself may further split (e.g., into one-dimensional subbundles with distinct Lyapunov exponents) (Hangyue, 30 Dec 2025, Mi et al., 12 Sep 2025).

A central dynamical hallmark is the assignment of definite sign Lyapunov exponents along $E^{cu}$ and $E^{cs}$ for all ergodic Gibbs $u$-states: the system is mostly expanding along $E^{cu}$ and mostly contracting along $E^{cs}$ (Hangyue, 30 Dec 2025).

2. Mixed Hyperbolicity and the $u$-definite Property

In the setting formalized by Mi–Cao (&&&3&&&), a key diagnostic is the $u$-definite property: for every ergodic Gibbs $u$-state (i.e., an invariant probability measure whose conditional measures along local strong-unstable leaves are absolutely continuous with respect to volume), all Lyapunov exponents along $E^c$ are of the same sign (either all positive or all negative). This leads to the canonical “mixed” regime: some physical measures have contracting central exponents, others have expanding central exponents, but individual measures never see both positive and negative exponents simultaneously in their center directions.

More generally, for splittings with multiple center subbundles, “mixed” refers to the situation where the Lyapunov exponents associated to different center subbundles may have opposite signs for different Gibbs $u$-states (e.g., one $E^c_i$ may be mostly expanding and another $E^c_j$ mostly contracting) (Mi et al., 12 Sep 2025).

3. Existence, Finiteness, and Stability of Physical Measures

A signature result for mixed partially hyperbolic systems is the existence and finiteness of SRB (Sinai–Ruelle–Bowen) or physical measures. Precisely, for a $C^2$ diffeomorphism $f$ with $u$-definite center bundle, there exist finitely many ergodic SRB measures $\mu_1,\dots,\mu_k$ whose basins cover a full Lebesgue measure subset of $M$ (Mi et al., 2023, Hangyue, 30 Dec 2025). Each SRB measure is a Gibbs $u$-state, and its basin

$$B(\mu_j) = \left\{ x \in M \mid \frac{1}{n}\sum_{i=0}^{n-1}\varphi(f^i(x)) \to \int\varphi\, d\mu_j, \ \forall \varphi \in C^0(M) \right\}$$

has positive Lebesgue measure. The system admits only finitely many physical measures, and their basins are essentially disjoint and collectively fill the ambient manifold modulo zero.

A major structural insight is that the number of physical measures is governed by the “skeleton” of the system—a finite set of hyperbolic periodic points controlling the supports of SRB measures. Under $C^1$ perturbations, the number of physical measures varies upper semi-continuously by the collapse or creation of skeleton points (e.g., merging of basins under creation of heteroclinic intersections) (Hangyue, 30 Dec 2025).

4. Canonical Examples and Geometric Constructions

Mixed partially hyperbolic systems arise in several geometric and model-theoretic contexts:

• Toral and nilmanifold automorphisms with center surgery: DA-type or Mañé-type modifications of toral automorphisms to break uniform hyperbolicity and create mixed centers (Carrasco et al., 2015).
• Nontrivial fiber bundles over simply connected manifolds: Fibered constructions with twisted torus factors over K3 or Kummer surfaces, using base diffeomorphisms with dominated splittings and nontrivial homotopy invariants in the bundle (Gogolev et al., 2014).
• Higher-dimensional toral examples: Systems with multi one-dimensional center bundles constructed via product maps and local surgery, yielding robust mixed examples with varying numbers and unstable indices of physical measures (Mi et al., 12 Sep 2025).
• Modifications of geodesic flows: Deformations of locally symmetric spaces of negative curvature yielding robustly partially hyperbolic but non-Anosov geodesic flows, with explicit mixed splitting and domination properties (Carneiro et al., 2011).
• Skew products over hyperbolic bases: Skew products of hyperbolic systems with smooth non-isometric (even weakly mixing) flows in the fiber can realize genuinely non-algebraic mixed partially hyperbolic systems with new Bernoulli examples (Dong et al., 2019).

5. Statistical and Ergodic Properties

Mixed partially hyperbolic systems exhibit rich ergodic behavior governed by the structure of their center bundles:

• Hierarchy of Gibbs $u$-states: All physical measures arise as Gibbs $u$-states; measures in which center exponents are negative satisfy mostly contracting conditions (e.g., Bonatti–Viana theory), while positive center exponents invoke “mostly expanding” theory (e.g., Alves–Bonatti–Viana).
• Entropy formulas and Ledrappier–Young theory: For Gibbs $cu$-states (ergodic invariant measures with positive exponents on $E^u\oplus E^c$), the entropy formula

$$h_\mu(f) = \int \log |\det Df|_{E^u\oplus E^c}|\,d\mu$$

holds and characterizes the equilibrium and absolute continuity properties necessary for SRB measures (Mi et al., 2023).

• Ergodic/Bernoulli properties: Some mixed partially hyperbolic examples are Bernoulli, including non-algebraic skew-products over Anosov bases with weakly mixing/slow-divergence central flows, expanding the repertoire of measure-theoretic models beyond algebraic settings (Dong et al., 2019).
• Persistence under perturbation: The dominated splitting and mixed Lyapunov profile are robust under $C^1$-small perturbations; the count of physical measures is upper semi-continuous (Hangyue, 30 Dec 2025).

6. Open Problems and Generalizations

Active directions and open problems include:

• Multiplicity and classification of skeletons: Extending explicit examples with more than two physical measures, classification of skeletons in higher center dimensions (Hangyue, 30 Dec 2025).
• Quantitative description of basins and mixing rates: Developing finer statistical descriptions, e.g., large deviation rates, speed of mixing, local limit theorems in the fast-slow non-uniform setting (Simoi et al., 2014).
• Statistical stability: Proving or quantifying continuity (or upper semi-continuity) of physical measures and their basins under smooth parameter variation in genuinely mixed regimes (Hangyue, 30 Dec 2025).
• Nontrivial topology: Exploring constructions on simply connected or exotic manifolds where Anosov systems never occur, yet mixed partial hyperbolicity is possible (Gogolev et al., 2014).

7. Summary Table: Properties of Representative Mixed Partially Hyperbolic Systems

Example Type	Center Structure	Physical Measure Properties
Kan-modified torus skew-product (Hangyue, 30 Dec 2025)	$E^{cu}$ mostly exp., $E^{cs}$ mostly ctr.	Two physical measures with distinct unstable indices; supports contain points of different indices
Torus with multi 1D centers (Mi et al., 12 Sep 2025)	$E^{c_1},E^{c_2},\ldots$ mixed, 1D each	$m$ physical measures, each with unique index, fixed under perturbation
Skew-product with quasi-elliptic fiber (Dong et al., 2019)	Center is smooth, non-isometric, weakly mixing	Bernoulli property, non-algebraic, center dynamics neither trivial nor isometric
Bernoulli examples on simply connected $6$-manifolds (Gogolev et al., 2014)	Center combines fiber and base neutral directions	Ergodic (Bernoulli), non-coherent examples, bundle structure essential
Partially hyperbolic geodesic flows (Carneiro et al., 2011)	Center generated by zero-curvature Jacobi fields	Robust non-Anosov, mixed splitting, not all measures fully hyperbolic

These examples illustrate the unifying framework of mixed partial hyperbolicity: robust domination, nonuniformly signed exponents in center directions, finite physical measure geometry, and stability features distinct from classical uniformly hyperbolic theory.

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References:

• (Mi et al., 2023) Physical measures for partially hyperbolic diffeomorphisms with mixed hyperbolicity
• (Hangyue, 30 Dec 2025) Variation of Physical Measures in Nontrivial Mixed Partially Hyperbolic Systems
• (Mi et al., 12 Sep 2025) Variety of physical measures in partially hyperbolic systems with multi 1-D centers
• (Dong et al., 2019) Bernoulli property for certain skew products over hyperbolic systems
• (Gogolev et al., 2014) New partially hyperbolic dynamical systems I
• (Carneiro et al., 2011) Partially hyperbolic geodesic flows
• (Carrasco et al., 2015) Partially hyperbolic dynamics in dimension 3
• (Simoi et al., 2014) Limit Theorems for Fast-slow partially hyperbolic systems